## Abstract

We study the pair correlation function for a variety of completely integrable quantum maps in one degree of freedom. For simplicity we assume that the classical phase space M is the Riemann sphere ℂP^{1} and that the classical map is a fixed-time map exptΞ_{H} of a Hamilton flow. The quantization is then a unitary N x N matrix U_{t,N}, and its pair correlation measure ρ^{(N)}_{2,} gives the distribution of spacings between eigenvalues in an interval of length comparable to the mean level spacing (∼ 1/N). The physicists' conjecture (Berry-Tabor conjecture) is that as N → ∞, ρ^{(N)}_{2,t} should converge to the pair correlation function ρ^{POISSON}_{2} = δ_{o} + 1 of a Poisson process. For any 2-parameter family of Hamiltonians of the form H_{α,β} = αφ(Î) + βÎ with φ″ ≠ 0 we prove that this conjecture is correct for almost all (α, β) along the subsequence of Planck constants N_{m} = [m(log m)^{5}]. In the addendum to this paper [Z. Addendum], we further show that for polynomial phases φ the a.e. convergence to Poisson holds along the full sequence of Planck constants for the Cesaro means of ρ^{(N)}_{2;(t,α,β)}.

Original language | English (US) |
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Pages (from-to) | 289-318 |

Number of pages | 30 |

Journal | Communications in Mathematical Physics |

Volume | 196 |

Issue number | 2 |

DOIs | |

State | Published - 1998 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics